if A≪Bmuch-less-thanABA\ll B, then there is C⊆XCXC\subseteq X, such that A≪C≪Bmuch-less-thanACmuch-less-thanBA\ll C\ll B.

By 1 and 4, it is easy to see that ∅≪∅much-less-than\varnothing\ll\varnothing. Also, 3 and 4 show that A∪C≪B∪Dmuch-less-thanACBDA\cup C\ll B\cup D whenever A≪Bmuch-less-thanABA\ll B and C≪Dmuch-less-thanCDC\ll D. So ≪much-less-than\ll is a topogenous order, which means≪much-less-than\ll is transitive and anti-symmetric. Under this orderrelation, we say that BBB is a proximal neighborhood of AAA if A≪Bmuch-less-thanABA\ll B.

The reason why we call BBB a “proximal” neighborhood is due to the following:

Theorem 1.

Let XXX be a set. The following are true.

Let ≪much-less-than\ll be defined as above. Define a new relation δδ\delta on P⁢(X)PXP(X): A⁢δ′⁢B′Asuperscriptδnormal-′superscriptBnormal-′A\delta^{{\prime}}B^{{\prime}} iff A≪Bmuch-less-thanABA\ll B. Then δδ\delta so defined is a proximity relation, turning XXX into a proximity space.

Conversely, let (X,δ)Xδ(X,\delta) is a proximity space. Define a new relation ≪much-less-than\ll on P⁢(X)PXP(X): A≪Bmuch-less-thanABA\ll B iff A⁢δ′⁢B′Asuperscriptδnormal-′superscriptBnormal-′A\delta^{{\prime}}B^{{\prime}}. Then ≪much-less-than\ll satisfies the six properties above.

Proof.

Suppose first that XXX and ≪much-less-than\ll are defined as above. We will verify the individual nearness relationaxioms of δδ\delta by proving their contrapositives in each case, except the last axiom:

1.

if A⁢δ′⁢BAsuperscriptδnormal-′BA\delta^{{\prime}}B, then A≪B′much-less-thanAsuperscriptBnormal-′A\ll B^{{\prime}}, or A⊆B′AsuperscriptBnormal-′A\subseteq B^{{\prime}}, so A∩B=∅ABA\cap B=\varnothing;

2.

suppose either A=∅AA=\varnothing or B=∅BB=\varnothing. In either case, A≪B′much-less-thanAsuperscriptBnormal-′A\ll B^{{\prime}}, which means A⁢δ′⁢BAsuperscriptδnormal-′BA\delta^{{\prime}}B;

3.

if A⁢δ′⁢BAsuperscriptδnormal-′BA\delta^{{\prime}}B, then A≪B′much-less-thanAsuperscriptBnormal-′A\ll B^{{\prime}}, so B′′≪A′much-less-thansuperscriptB′′superscriptAnormal-′B^{{\prime\prime}}\ll A^{{\prime}}, or B≪A′much-less-thanBsuperscriptAnormal-′B\ll A^{{\prime}}, or B⁢δ′⁢ABsuperscriptδnormal-′AB\delta^{{\prime}}A;

4.

if A1⁢δ′⁢BsubscriptA1superscriptδnormal-′BA_{1}\delta^{{\prime}}B and A2⁢δ′⁢BsubscriptA2superscriptδnormal-′BA_{2}\delta^{{\prime}}B, then A1≪Bmuch-less-thansubscriptA1BA_{1}\ll B and A2≪Bmuch-less-thansubscriptA2BA_{2}\ll B, so (A1∪A2)≪Bmuch-less-thansubscriptA1subscriptA2B(A_{1}\cup A_{2})\ll B, or (A1∪A2)⁢δ′⁢BsubscriptA1subscriptA2superscriptδnormal-′B(A_{1}\cup A_{2})\delta^{{\prime}}B;

5.

if A⁢δ′⁢BAsuperscriptδnormal-′BA\delta^{{\prime}}B, then A≪B′much-less-thanAsuperscriptBnormal-′A\ll B^{{\prime}}. So there is D⊆XDXD\subseteq X with A≪Dmuch-less-thanADA\ll D and D≪B′much-less-thanDsuperscriptBnormal-′D\ll B^{{\prime}}. Let C=D′CsuperscriptDnormal-′C=D^{{\prime}}. Then A≪C′much-less-thanAsuperscriptCnormal-′A\ll C^{{\prime}} and C′≪B′much-less-thansuperscriptCnormal-′superscriptBnormal-′C^{{\prime}}\ll B^{{\prime}}, or A⁢δ′⁢CAsuperscriptδnormal-′CA\delta^{{\prime}}C and C′⁢δ′⁢BsuperscriptCnormal-′superscriptδnormal-′BC^{{\prime}}\delta^{{\prime}}B.

since X⁢δ′⁢∅Xsuperscriptδnormal-′X\delta^{{\prime}}\varnothing, X≪∅′much-less-thanXsuperscriptnormal-′X\ll\varnothing^{{\prime}}, or X≪Xmuch-less-thanXXX\ll X;

2.

suppose A⁢δ′⁢B′Asuperscriptδnormal-′superscriptBnormal-′A\delta^{{\prime}}B^{{\prime}}, then if x∈AxAx\in A, we have x⁢δ′⁢B′xsuperscriptδnormal-′superscriptBnormal-′x\delta^{{\prime}}B^{{\prime}}, implying x∩B′=∅xsuperscriptBnormal-′x\cap B^{{\prime}}=\varnothing, or x∈BxBx\in B;

3.

if A≪Bmuch-less-thanABA\ll B and C≪Dmuch-less-thanCDC\ll D, then A⁢δ′⁢B′Asuperscriptδnormal-′superscriptBnormal-′A\delta^{{\prime}}B^{{\prime}} and C⁢δ′⁢D′Csuperscriptδnormal-′superscriptDnormal-′C\delta^{{\prime}}D^{{\prime}}, which means A⁢δ′⁢(B′∪D′)Asuperscriptδnormal-′superscriptBnormal-′superscriptDnormal-′A\delta^{{\prime}}(B^{{\prime}}\cup D^{{\prime}}) and C⁢δ′⁢(B′∪D′)Csuperscriptδnormal-′superscriptBnormal-′superscriptDnormal-′C\delta^{{\prime}}(B^{{\prime}}\cup D^{{\prime}}), which together imply (A∩C)⁢δ′⁢(B′∪D′)ACsuperscriptδnormal-′superscriptBnormal-′superscriptDnormal-′(A\cap C)\delta^{{\prime}}(B^{{\prime}}\cup D^{{\prime}}), or (A∩C)⁢δ⁢(B∩D)′ACδsuperscriptBDnormal-′(A\cap C)\delta(B\cap D)^{{\prime}}, or A∩C≪B∩Dmuch-less-thanACBDA\cap C\ll B\cap D;

4.

if A≪Bmuch-less-thanABA\ll B, then A⁢δ′⁢B′Asuperscriptδnormal-′superscriptBnormal-′A\delta^{{\prime}}B^{{\prime}}, so B′⁢δ′⁢AsuperscriptBnormal-′superscriptδnormal-′AB^{{\prime}}\delta^{{\prime}}A (as δδ\delta is symmetric, so is its complement), which is the same as B′⁢δ′⁢A′′superscriptBnormal-′superscriptδnormal-′superscriptA′′B^{{\prime}}\delta^{{\prime}}A^{{\prime\prime}}, or B′≪A′much-less-thansuperscriptBnormal-′superscriptAnormal-′B^{{\prime}}\ll A^{{\prime}};

5.

if A⁢δ⁢D′AδsuperscriptDnormal-′A\delta D^{{\prime}}, then B⁢δ⁢C′BδsuperscriptCnormal-′B\delta C^{{\prime}} (since A⊆BABA\subseteq B and D′⊆C′superscriptDnormal-′superscriptCnormal-′D^{{\prime}}\subseteq C^{{\prime}}), so B≪′Csuperscriptmuch-less-thannormal-′BCB\ll^{{\prime}}C, a contradiction;

6.

if A≪Bmuch-less-thanABA\ll B, then A⁢δ′⁢B′Asuperscriptδnormal-′superscriptBnormal-′A\delta^{{\prime}}B^{{\prime}}, so there is D⊆XDXD\subseteq X with A⁢δ′⁢DAsuperscriptδnormal-′DA\delta^{{\prime}}D and D′⁢δ′⁢B′superscriptDnormal-′superscriptδnormal-′superscriptBnormal-′D^{{\prime}}\delta^{{\prime}}B^{{\prime}}. Define C=D′CsuperscriptDnormal-′C=D^{{\prime}}, then A≪Cmuch-less-thanACA\ll C and C≪Bmuch-less-thanCBC\ll B, as desired.

Because of the above, we see that a proximity space can be equivalently defined using the proximal neighborhood concept. To emphasize its relationship with δδ\delta, a proximal neighborhood is also called a δδ\delta-neighbhorhood.

Furthermore, we have

Theorem 2.

if BBB is a proximal neighborhood of AAA in a proximity space (X,δ)Xδ(X,\delta), then BBB is a (topological) neighborhood of AAA under the topology τ⁢(δ)τδ\tau(\delta) induced by the proximity relation δδ\delta. In other words, if A≪Bmuch-less-thanABA\ll B, then A⊆B∘AsuperscriptBA\subseteq B^{{\circ}} and Ac⊆BsuperscriptAcBA^{c}\subseteq B, where ∘{}^{{\circ}} and cc{}^{c} denote the interior and closure operators.

Proof.

Since A⁢δ′⁢B′Asuperscriptδnormal-′superscriptBnormal-′A\delta^{{\prime}}B^{{\prime}}, then x⁢δ′⁢B′xsuperscriptδnormal-′superscriptBnormal-′x\delta^{{\prime}}B^{{\prime}} whenever x∈AxAx\in A, which is the contrapositive of the statement: x∈A′xsuperscriptAnormal-′x\in A^{{\prime}} whenever x⁢δ⁢B′xδsuperscriptBnormal-′x\delta B^{{\prime}}, which is equivalent to B′⁣c⊆A′superscriptBnormal-′csuperscriptAnormal-′B^{{\prime c}}\subseteq A^{{\prime}}, or A⊆B∘AsuperscriptBA\subseteq B^{{\circ}}. Furthermore, if x∉BxBx\notin B, then x∈B′xsuperscriptBnormal-′x\in B^{{\prime}}. But A⁢δ′⁢B′Asuperscriptδnormal-′superscriptBnormal-′A\delta^{{\prime}}B^{{\prime}} b assumption. This implies x⁢δ′⁢Axsuperscriptδnormal-′Ax\delta^{{\prime}}A, which means x∉AcxsuperscriptAcx\notin A^{c}. Therefore Ac⊆BsuperscriptAcBA^{c}\subseteq B.
∎

Remark. However, not every τ⁢(δ)τδ\tau(\delta)-neighborhood is a δδ\delta-neighborhood.